I dont really think your team should NEED to have specific moves to win.

It just needs alot of synergy. Even if the pets aren't that good, if they have synergy that will generally win.

Take for example Infected Fawn and Son of Animus. You'd generally think "Oh, easy win."

Well, its actually not.
[ability]Touch of the Animus[/ability] + [ability]Death and Decay[/ability]/[ability]Flurry[/ability] means you'll sure be getting some health!

This goes for all pets and teams. Having no synergy can really hurt you.

You also want to find if your team has a general family its weak to. Take for example this team:

[pet]Snarly[/pet]
[pet]Scalded Basilisk Hatchling[/pet]

What family is this team weak to? You guessed it! FLYING!

Snarly takes increased damage from it while the Basilisk's [ability]Thrash[/ability] is weak against Flying AND is not able to get the additional hit from it's ability.
So you want to put a double counter. A double counter pet is a pet that can absolutely dominate a certain family. In this case, we want to dominate flying. So we pick the [pet]Emerald Proto-Whelp[/pet]. The whelp can both do increased damage to flyers with its [ability]Emerald Bite[/ability] but also takes reduced damage from flyers.

Going to second that part about synergy. I have a team based around the 'Chilled' effect and I've had some great success with (surprisingly). None of the three pets are really that special, but when you have three pets that can greatly benefit from that specific weather condition and two pets that can control the weather, you can actually compensate a lot for weak individual pets or even a lack of type variety.